Definition:Indexing Set/Family

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Definition

Let $I$ and $S$ be sets.

Let $x: I \to S$ be a mapping.

Let the domain $I$ of $x$ be the indexing set of $\left \langle {x_i} \right \rangle_{i \in I}$.

The mapping $x$ itself is called a family of elements of $S$ indexed by $I$.

The object $\left \langle {x_i} \right \rangle_{i \in I}$ is often referred to as an $I$-indexed family.

The family $x$ is often seen with one of the following notations:

$\left \langle {x_i} \right \rangle_{i \in I}$

$\left({x_i}\right)_{i \in I}$

$\left\{{x_i}\right\}_{i \in I}$

There is little consistency in the literature.

The subscripted $i \in I$ is often left out, if it is obvious in the particular context.